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Holographic Theory, and Emergent Gravity As an Entropic Force

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Holographic Theory, and Emergent Gravity As an Entropic Force Holographic theory, and emergent gravity as an entropic force Liu Tao December 9, 2020 Abstract In this report, we are going to study the holographic principle, and the possibility of describing gravity as an entropic force, independent of the details of the microscopic dynamics. We will first study some of the deep connections between Gravity laws and thermodynamics, which motivate us to consider gravity as an entropic force. Lastly, we will investigate some theoretical criticisms and observational evidences on emergent gravity. 1 Contents 1 Introduction3 2 Black holes thermodynamics and the holographic principle3 2.1 Hawking temperature, entropy..........................3 2.1.1 Uniformly accelerated observer and the Unruh effect.........3 2.1.2 Hawking temperature and black hole entropy..............6 2.2 Holographic principle...............................7 3 Connection between gravity and thermodynamics8 3.1 Motivations (from ideal gas and GR near event horizon)...........8 3.2 Derive Einstein's law of gravity from the entropy formula and thermodynamics 10 3.2.1 the Raychaudhuri equation........................ 10 3.2.2 Einstein's equation from entropy formula and thermodynamics.... 11 4 Gravity as an entropic force 13 4.1 General philosophy................................ 13 4.2 Entropic force in general............................. 13 4.3 Newton's law of gravity as an entropic force.................. 14 4.4 Einstein's law of gravity as an entropic force.................. 15 4.5 New interpretations on inertia, acceleration and the equivalence principle.. 17 5 Theoretical and experimental criticism 18 5.1 Theoretical criticism............................... 18 5.2 Cosmological observations............................ 19 20section.6 2 1 Introduction The holographic principle was inspired by black hole thermodynamics, which conjectures that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected for any extensive quantities. We will start with a brief introduction on black hole thermodynamics in Section 2 Black holes thermodynamics and the holographic principle where we address the Hawking temperature and the black hole entropy formula, also known as the Bekenstein{Hawking formula. We then investigate some deep connec- tions between Gravity laws and thermodynamics in Section 3 Connection between gravity and thermodynamics. We first consider two examples from ideal gas and Einstein's field equations near the black hole event horizon as motivations [1,2]. Then we derive the Ein- stein's law of gravity from the entropy formula and the law of thermodynamics in all its glory following [1]. Afterwards, in Section 4 Gravity as an entropic force we follow [3] and explain gravity as an entropic force generated by changes in the information associated with the positions of material bodies in both non-relativistic and relativistic settings. With this logic in mind we recover the Newton's law of gravity in a simple non-relativistic setting and Einstein's field equation when we generalize these results to the relativistic case. What's more, when space is emergent even Newton's law of inertia and and Einstein's equivalence principle need to be reexplained. So we will give new interpretations on Newton's law of inertia and the meaning of acceleration (both concepts in the original theories are related to spacetime-specific notions such as coordinates) in this new emergent space scenario where only the fundamental concepts such as entropy and temperature are essential. The equiva- lence principle can also be explained more naturally and unavoidably in a theory in which space is emergent through a holographic scenario since both gravity and acceleration are emergent phenomena. Last but not least, we summarize some of the theoretical criticism and observational evidences about emergent gravity theory of [3] in Section 5 Theoretical and experimental criticism. Despite all of the theoretical evidences presented in this report, there is a decreasing research interest into the emergent gravity theories within the community in recent years as far as I can tell. Maybe the last section can provide some clues. We will use the mostly pluses signature (−; +; +; +) throughout and units with kB = c = h¯ = 1.1 2 Black holes thermodynamics and the holographic prin- ciple 2.1 Hawking temperature, entropy 2.1.1 Uniformly accelerated observer and the Unruh effect We are going to show in explicit details that for a uniformly accelerated observer [4], the ground state of an inertial observer is seen as a mixed state in thermodynamic equilibrium 1for the most part, since sometimes havingh ¯ in our result explicitly for example can emphasize the its quantum nature. 3 with a non-zero temperature bath, and the observed frequency spectrum is given by familiar Planck's law with the temperature T = 1/β = a=(2π), where a is the constant acceleration. In the reference of a stationary observer, the four-acceleration of our uniformly accelerated observer is aα =x ¨α = (0; a), where jaj = a is constant. We can convert this condition into a covariant form α β 2 ηαβx¨ x¨ = a (1) we also have the normalization condition α β ηαβx_ x_ = −1 (2) Without lose generality, let's assume that the trajectory is contained in the t{x plane, i.e. a = (a; 0; 0). Switching to the light-cone coordinates u = t − x and v = t + x (3) the line elements can be written as (suppress the transverse coordinates y and z) ds2 = −dt2 + dx2 = −dudv (4) The acceleration equation1 and the normalization equation2 can be written as u¨v¨ = −a2 andu _v_ = 1 (5) The two equations in5 can be combined to eliminate one of the variable say u, we obtain v_ = ±a (6) v_ This is integrated to be A v(τ) = exp(aτ) + C (7) a and, usingu _ = 1=v_ we got 1 u(τ) = − exp(−aτ) + D (8) Aa go back to the original Cartesian coordinates, we obtain2 1 1 t(τ) = sinh(aτ) and x(τ) = cosh(aτ) (9) a a where we have chosen the integration constants such that the initial conditions t(0) = 0 and x(0) = 1=a are satisfied. Now consider a monochromatic wave for the inertial observer φ(k) / exp(−i!(t − x)) (10) 2This result is also in one of our homework questions. 4 An accelerated observer does not see a monochromatic wave however, but a superposition of plane waves with varying frequencies. We can see this by inserting the trajectory9 into the monochromatic wave i! i! φ(τ) / exp − [sinh(aτ) − cosh(aτ)] = exp exp(−aτ) (11) a a As the next step, we want to determine its power spectrum P (ν) = jφ(ν)j2 and relate it to the Plank's formula. First let us calculate the wave 11 in the Fourier space φ(ν). According to the Fourier transform, we obtain Z 1 Z 1 i! φ(ν) = dτφ(τ)e−iντ = dτ exp exp(−aτ) e−iντ (12) −∞ −∞ a change variable to y = exp(−aτ), we obtain 1 Z 1 φ(ν) = dyyiν=a−1ei(!=a)y (13) a 0 we notice the above integral is similar to the Gamma function integral Z 1 dttz−1e−bt = b−zΓ(z) = exp(−z ln b)Γ(z) (14) 0 except our integral 13 is not convergent. Here we added an infinitesimal positive real quantity " to ensure the convergence. Combine equations 13 and 14, we obtain 1 φ(ν) = · exp (−iν=a ln(−i!=a + )) · Γ(iν=a) (15) a i! ! iπ where we also have lim"!0 ln − a + " = ln a − 2 sign(!=a). Use this we obtain 1 ! −iν=a φ(ν) = Γ(iν=a)e−πν=(2a) (16) a a for positive !=a. Thus, the power spectrum is 1 P (ν) = jφ(ν)j2 = e−πν=aΓ(iν=a)Γ(−iν=a) (17) a2 use Euler's reflection formula −π πa Γ(iν=a)Γ(−iν=a) = iν ν = πν (18) a · sin(iπ a ) ν sinh a equation 17 now becomes π e−πν=a β 1 P (ν) = = (19) a2 (ν=a) sinh(πν=a) ν eβν − 1 with β = 2π=a. This is the Planck's formula. Therefore, classically, a uniformly accelerated detector will measure a thermal Planck spectrum with temperature T = 1/β = a=(2π). This phenomenon is called Unruh effect [5]. 5 This result can also be easily generalized to quantum field theory. Quantum mechanically, in the canonical quantization formalism3, we can follow the same steps and consider instead any quantum fields with harmonic expansions similar to the classical monochromatic wave we considered above, but with integration over all available frequency modes. As a result, the vacuum expectation value of the number density operator of scalar particles detected by a uniformly accelerated observer for example is [4] 1 hn~ i = (20) Ω exp(2πΩ=a) − 1 which is again, the black body radiation with the temperature given by T = 1/β = a=(2π). In special relativity, an observer moving with uniform proper acceleration through Minkowski spacetime is conveniently described with Rindler coordinates, which are related to the stan- dard (Cartesian) Minkowski coordinates by x = ρ cosh(σ) (21) t = ρ sinh(σ) The line element in Rindler coordinates is ds2 = −ρ2dσ2 + dρ2 (22) 1 where ρ = a and σ = aτ for an uniformly accelerated observer, see equation9. In the next section where we derive the black hole entropy, we are going to massage the Schwarzschild metric outside a black hole into the above 22 form and interpret the resulting Unruh temperature of stationary observer at the spatial infinity as the well-known Hawking temperature. Combine the Hawking temperature and the second law of thermodynamics we can finally derive the black hole entropy formula. 2.1.2 Hawking temperature and black hole entropy As briefly outlined above, in order to calculate the black hole entropy, let's start with the Schwarzschild metric [2,7] dr2 ds2 = −f(r)dt2 + + r2 dθ2 + sin2 θdφ2 (23) f(r) where f(r) ≡ 1 − 2GM=r.
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